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Hippasus, Philolaus, and Eudoxus. Pythagoras (580-500? B.C.) conceived mathematics to
be the most sacred and exact of all the sciences, and demanded of all who came to him
for study a familiarity with arithmetic, music, astronomy, and geometry. He laid special
emphasis upon the philosophic life as a prerequisite to wisdom. Pythagoras was one of
the first teachers to establish a community wherein all the members were of mutual
assistance to one another in the common attainment of the higher sciences. He also
introduced the discipline of retrospection as essential to the development of the spiritual
mind. Pythagoreanism may be summarized as a system of metaphysical speculation
concerning the relationships between numbers and the causal agencies of existence. This
school also first expounded the theory of celestial harmonics or "the music of the
spheres." John Reuchlin said of Pythagoras that he taught nothing to his disciples before
the discipline of silence, silence being the first rudiment of contemplation. In his Sophist,
Aristotle credits Empedocles with the discovery of rhetoric. Both Pythagoras and
Empedocles accepted the theory of transmigration, the latter saying: "A boy I was, then
did a maid become; a plant, bird, fish, and in the vast sea swum." Archytas is credited
with invention of the screw and the crane. Pleasure he declared to be a pestilence because
it was opposed to the temperance of the mind; he considered a man without deceit to be
as rare as a fish without bones.

The Eleatic sect was founded by Xenophanes (570-480 B.C.), who was conspicuous for
his attacks upon the cosmologic and theogonic fables of Homer and Hesiod. Xenophanes
declared that God was "one and incorporeal, in substance and figure round, in no way
resembling man; that He is all sight and all hearing, but breathes not; that He is all things,
the mind and wisdom, not generate but eternal, impassible, immutable, and rational."
Xenophanes believed that all existing things were eternal, that the world was without
beginning or end, and that everything which was generated was subject to corruption. He
lived to great age and is said to have buried his sons with his own hands. Parmenides
studied under Xenophanes, but never entirely subscribed to his doctrines. Parmenides
declared the senses to be uncertain and reason the only criterion of truth. He first asserted
the earth to be round and also divided its surface into zones of hear and cold.

Melissus, who is included in the Eleatic school, held many opinions in common with
Parmenides. He declared the universe to be immovable because, occupying all space,
there was no place to which it could be moved. He further rejected the theory of a
vacuum in space. Zeno of Elea also maintained that a vacuum could not exist. Rejecting
the theory of motion, he asserted that there was but one God, who was an eternal,
ungenerated Being. Like Xenophanes, he conceived Deity to be spherical in shape.
Leucippus held the Universe to consist of two parts: one full and the other a vacuum.
From the Infinite a host of minute fragmentary bodies descended into the vacuum, where,
through continual agitation, they organized themselves into spheres of substance.

The great Democritus to a certain degree enlarged upon the atomic theory of Leucippus.
Democritus declared the principles of all things to be twofold: atoms and vacuum. Both,
he asserted, are infinite--atoms in number, vacuum in magnitude. Thus all bodies must be
composed of atoms or vacuum. Atoms possessed two properties, form and size, both
characterized by infinite variety. The soul Democritus also conceived to be atomic in